On Positively Quadratically Hyponormal Weighted Shifts

Abstract

Consider the sequence of positive weights α(x, y) : √ x, √ y,

3 4 ,

4 5 , . . . with a Bergman tail. If y = 2 3 then it was shown in [2] that for 0 < x ≤ y, the weighted shift operatorWα(x,y) is positively quadratically hyponormal. In this paper we show that there exists an interval (k1, k2) about 2 3 such that if y ∈ (k1, k2) then for 0 < x ≤ y ,Wα(x,y) is positively quadratically hyponormal. In fact, using Mathematica graphs we show that the largest such interval is [k1, k2) where k1 = 29 46 ≈ 0.630435 and k2 = 0.737144.